
688 SECOND-ORDER HYPERBOLIC EQUATIONS WITH TWO SPACE VARIABLES
Here,
G(r, ϕ, ξ, η, t) = exp
−
1
2
kt
∞
X
n=0
∞
X
m=1
A
nm
J
s
n
(µ
nm
r)J
s
n
(µ
nm
ξ) cos(s
n
ϕ) cos(s
n
η) sin
λ
nm
t
,
s
n
=
nπ
ϕ
0
, A
nm
=
4µ
2
nm
ϕ
0
(µ
2
nm
R
2
+ β
2
R
2
− s
2
n
)
J
s
n
(µ
nm
R)
2
λ
nm
, λ
nm
=
q
a
2
µ
2
nm
+ b −
1
4
k
2
,
where the J
s
n
(r) are Bessel functions and the µ
nm
are positive roots of the transcendental
equation
µJ
′
s
n
(µR) + βJ
s
n
(µR) = 0.
7.4.3 A xisymmetric Problems
In the axisymmetric case, a nonhomogeneous telegraph equation in the cylindrical coordi-
nate system has the form
∂
2
w
∂t
2
+ k
∂w
∂t
= a
2
∂
2
w
∂r
2
+
1
r
∂w
∂r
+
∂
2
w
∂z
2
− bw + Φ(r, z, t), r =
p
x
2
+ y
2
.
In the solutions of the problems considered below, the modified Green’s function
G(r,